Approximation of biholomorphic maps between Runge domains by holomorphic automorphisms

TL;DR

The paper proves that biholomorphic maps between specific Runge domains in complex space can be approximated by holomorphic automorphisms, extending known results and simplifying proofs in complex analysis.

Contribution

It introduces a new approximation result for biholomorphic maps between Runge domains using holomorphic automorphisms, generalizing previous work and applying to Stein manifolds.

Findings

Biholomorphic maps are limits of automorphisms in certain Runge domains

Results extend to volume-preserving maps and Stein manifolds with density property

Provides a simpler proof for approximation of biholomorphic maps

Abstract

We show that biholomorphic maps between certain pairs of Runge domains in the complex affine space $\mathbb C^n$, $n>1$, are limits of holomorphic automorphisms of $\mathbb C^n$. A similar result holds for volume preserving maps and also in Stein manifolds with the density property. This generalizes several results in the literature and provides a considerably simpler proof.